In 1973, Fischer Black and Myron Scholes published their groundbreaking paper “the pricing of options and corporate liabilities”. Not only did this specify the first successful options pricing formula, but it also described a general framework for pricing other derivative instruments. That paper launched the field of financial engineering. Black and Scholes had a hard time getting that paper published. Eventually, it took intercession by Eugene Fama and Merton Miller to get it accepted by the *Journal of Political Economy*. In the mean time, Black and Scholes had published in the *Journal of Finance* a more accessible (1972) paper that cited the as-yet unpublished (1973) option pricing formula in an empirical analysis of current options trading.

The Black-Scholes (1973) option pricing formula prices European put or call options on stocks. It assumes the underlying stock price follows a geometric Brownian motion with constant volatility. It further assumes the stock does not pay a dividend or make other distributions. While the Black-Scholes (1973) option pricing formula is historically important, that last assumption limits its practical applicability.

Values for a call price *c* or put price *p* are:

[1]

[2]

where:

[3]

[4]

Here, *log* denotes the natural logarithm, and:

*s*= the price of the underlying stock*x*= the strike price*r*= the continuously compounded risk free interest rate*t*= the time in years until the expiration of the option- σ = the implied volatility for the underlying stock
- Φ = the standard normal cumulative distribution function.

Consider a European call option on 100 shares of non-dividend-paying stock ABC. The option is struck at USD 55 and expires in .34 years. ABC is trading at USD 56.25 and has 28% (that is .28) implied volatility. The continuously compounded risk free interest rate is .0285. Applying formula [1], the option’s market value per share of ABC is USD 4.56. Since the call is for 100 shares, its total value is USD 456. Of this, USD 125 is intrinsic value, and USD 331 is time value.

The Greeks—delta, gamma, vega, theta and rho—for a call are:

[5]

[6]

[7]

[8]

[9]

where ϕ denotes the standard normal probability density function. For a put, the Greeks are:

[10]

[11]

[12]

[13]

[14]

Note that gamma formulas [6] and [11] are identical for puts and calls, as are vega formulas [7] and [12].

## References

- Black, Fischer and Myron S. Scholes (1972) The valuation of option contracts and a test of market efficiency,
*Journal of Finance*, 27 (2), 399–418. - Black, Fischer and Myron S. Scholes (1973). The pricing of options and corporate liabilities,
*Journal of Political Economy*, 81 (3), 637-654.